minor corrections
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@ 619,7 +619,7 @@ The timedependent Green's matrix, Eq.~\eqref{eq:cn}, is then rewritten as


\prod_{k=0}^{N1} \mel{ i_k }{ T }{ i_{k+1} } .


\ee


$\cC_p$ can be partitioned further by considering the subset of paths, denoted by $\cC_p^{I,n}$,


having $(\ket{I_k}, n_k)$ for $k=0$ to $p$ as exit states and trapping times.


having $(\ket{I_k}, n_k)$, for $0 \le k \le p$, as exit states and trapping times.


We recall that, by definition of the exit states, $\ket{I_k} \notin \cD_{I_{k1}}$. The total time must be conserved, so the relation $\sum_{k=0}^p n_k= N+1$


must be fulfilled.


Now, the contribution of $\cC_p^{I,n}$ to the path integral is obtained by summing over all elementary paths $(i_1,...,i_{N1})$


@ 1009,7 +1009,7 @@ The two variational parameters of the trial vector have been optimized and fixed


variational energy $E_\text{v}=0.495361\ldots$.


In what follows, $\ket{I_0}$ is systematically chosen as one of the two N\'eel states, \eg, $\ket{I_0} = \ket{\uparrow,\downarrow, \uparrow,\ldots}$.




Figure \ref{fig3} shows the convergence of $H^{DMC}_p$, Eq.~\eqref{eq:defHp}, as a function of $p$ for different values of the reference energy $E$.


Figure \ref{fig3} shows the convergence of $H^\text{DMC}_p$, Eq.~\eqref{eq:defHp}, as a function of $p$ for different values of the reference energy $E$.


We consider the simplest case where a singlestate domain is associated to each state.


Five different values of $E$ have been chosen, namely $E=1.6$, $1.2$, $1.0$, $0.9$, and $0.8$.


Only $H_0$ is computed analytically ($p_\text{ex}=0$).


@ 1129,7 +1129,7 @@ The extrapolated values obtained from the five values of the energy with three d


Fitting the data using a simple linear function leads to an energy of $0.7680282(5)$ (to be compared with the exact value of $0.768068\ldots$).


A small bias of about $4 \times 10^{5}$ is observed.


This bias vanishes within the statistical error when resorting to more flexible fitting functions, such as a quadratic function of $E$ or the


twocomponent representation given by Eq.(\ref{eq:2comp}).


twocomponent representation given by Eq.~\eqref{eq:2comp}.


Our final value is in full agreement with the exact value with about six decimal places.




Table \ref{tab4} shows the evolution of the average trapping times and extrapolated energies as a function of $U$ when using $\cD(0,1) \cup \cD(1,0)$ as the main domain.


@ 1154,7 +1154,7 @@ All these aspects will be considered in a forthcoming work.


%%% TABLE II %%%


\begin{table}


\caption{Onedimensional Hubbard model with $N=4$, $U=12$, and $E=1$.


Dependence of the statistical error on the energy with the number of $p$components $p_{ex}$ calculated analytically in the expression


Dependence of the statistical error on the energy with the number of $p$components $p_\text{ex}$ calculated analytically in the expression


of the energy, $\cE^\text{DMC}(E,p_\text{ex},p_\text{max})$, Eq.~\eqref{eq:E_pex}.


The simulation is performed the same way as in Table \ref{tab1}.


Results are presented when a singlestate domain is used for all states and when $\cD(0,1) \cup \cD(1,0)$ is chosen as the main domain.}



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